Pricing high-dimensional American options by kernel ridge regression

Abstract

In this paper, we propose using kernel ridge regression (KRR) to avoid the step of selecting basis functions for regression-based approaches in pricing high-dimensional American options by simulation. Our contribution is threefold. Firstly, we systematically introduce the main idea and theory of KRR and apply it to American option pricing for the first time. Secondly, we show how to use KRR with the Gaussian kernel in the regression-later method and give the computationally efficient formulas for estimating the continuation values and the Greeks. Thirdly, we propose to accelerate and improve the accuracy of KRR by performing local regression based on the bundling technique. The numerical test results show that our method is robust and has both higher accuracy and efficiency than the Least Squares Monte Carlo method in pricing high-dimensional American options.

Towards control of dam and reservoir systems with forward–backward stochastic differential equations driven by clustered jumps

“…Therefore, the choice will work well only for the former test case. The finding that smaller is better for larger number of sample paths is consistent with the existing results 78 …”

“…However, this will not be always the case. 78 There would exist some optimal scaling relationship between the total numbers of sample paths and bundles, which is possibly problem-dependent and should be addressed with deeper stochastic analysis. In this article, the stochastic grid bundling has been applied only to the space of inflow, which is an uncontrolled and is therefore not updated at each Picard iteration.…”

“…Therefore, sharper computational results having smaller deviations would be derived with a larger total number of bundles. However, this will not be always the case 78 . There would exist some optimal scaling relationship between the total numbers of sample paths and bundles, which is possibly problem‐dependent and should be addressed with deeper stochastic analysis.…”

An Iteration Algorithm for American Options Pricing Based on Reinforcement Learning

“…Let {X t , 0 ≤ t ≤ T} be a R d -valued Markov process, this process is defined on a filtered measurable probability space with a risk-neutral measure P. We assumed that the process records all relevant financial variables. In practice, the price of the American option was approximated by the price of a Bermudan option [11], which could be exercised at discrete time points 0 < t 1 < • • • < t N = T. For 0 ≤ n ≤ N, we represented t n by n to simplify the notation in the following. We assumed that the risk-free discount factor between time points was constant, which was denoted by γ ∈ (0, 1).…”

“…The most popular are the regression-based methods proposed by Longstaff and Schwartz [7] and Tsitsiklis and Roy [8]. Through a backward iteration scheme, these methods can approximate the continuation value and a feasible exercise policy, such as linear regression [7], neural network [9], Gaussian process regression [10] and kernel ridge regression [11], all of which produce lower price bounds for American options. The dual approaches for American options were developed by Rogers [12] and Haugh and Kogan [13], these methods produce upper price bounds for options.…”